The proposed research is in geometric evolution equations, embedding problems, analysis on metric spaces, and geometric group theory. The evolution equations in the proposal are mean curvature flow and Ricci flow. The proposed research in analysis on metric spaces clusters in three areas: (1) bilipschitz embedding problems and related issues, (2) harmonic functions, isometry groups, and asymptotic structure of doubling spaces (e.g. manifolds with nonnegative Ricci curvature), (3) the structure of boundaries of Gromov hyperbolic spaces. Common themes in all three areas are spaces satisfying Poincare inequalities, and rescaling arguments leading to singular limit spaces.
The project aims to study two nonlinear analogs of the heat equation: evolution of surfaces by mean curvature, and Hamilton's Ricci flow. Evolution by mean curvature has been studied for decades as a natural model for evolving surface interfaces. Ricci flow describes an evolving geometry, and was used in Perelman's solution of the Poincare conjecture. The primary objective of the proposed research on these equations is to study singularities and show that they have a very special form. Another component of the research program is an investigation of spaces which have a self-similar or fractal character, using analytic tools that have been developed in the last few years. Here one of the goals is to deform the space into an optimal form, if possible, in order to reveal hidden symmetries, and otherwise show that no hidden symmetries exist. This is very useful for understanding the asymptotic shape of infinite groups, and is part of confluence of several research trends over the last 10-15 years. Another application of similar ideas is to embedding problems in theoretical computer science: Cheeger, Naor, and the PI were able to substantially improve the previous best known results on the embedding of spaces of negative type, in connection with the quantitative version of the Goemans-Linial conjecture.